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Question 63

If a straight lines moves so that the sum of the perpendiculars let fall on it from the fixed points (3, 4) and (7, 2) is equal to three times the perpendicular on it from a fixed point (1, 3), prove that there is another fixed point through which the line always passes and find its coordinates.

My Self Assessment

   CLEAR

Prove that the equations r = acos (θ–α) and r = bsin (θ – α) represent two circles which cut at right angles.

Prove that the equation r2cosθ – arcos2θ – 2a2cosθ = 0 represents a straight line and a circle.

Find the condition that the straight line  may touch the circle r = 2c cosθ

a) a2b2 + 2ac = 1

b) a2b2 + 2bc = 1

c) b2c2 + 2ac = 1

d) b2 + c2 = 1

A point moves so that the sum of the square of its distances from the four sides of a square is constant. Prove that it always lies on a circle.

A point moves so that the sum of the squares of its distances from the angular points of a triangle is constant. Prove that its locus is a circle.

Find the locus of a point which moves so that the square of the tangents drawn from it to the circle x2 + y2 = a2 is equal to c times the distance from the straight line lx + my + n = 0

a) Circle

b) Ellipse

c) Parabola

d) None of the above

Find the locus of a point whose distance from a fixed point is in a constant ratio to the tangent drawn from it to a circle.

a) Circle

b) Parabola

c) Ellipse

d) Hyperbola

Find the locus of the vertex of a triangle, given its base and the sum of the squares of sides

Find the locus of the vertex of a triangle, given its base and the sum of m times the square of one side and n times the square of the other side.

A point moves so that sum of the squares of its distances from n fixed points is given. Prove that its locus is a circle.

Whatever the value of α, prove that the locus of the intersection of the straight lines x cosα + y sinα = a and x sinα – y cosα = b is a circle.

From a point P on a circle perpendiculars PM and PN are drawn to two radii of a circle which are at right angles; find the locus of middle point of MN.

Find the locus of the middle point of chords of the circle x2 + y2 = a2 which pass through the point (h, k).

A straight line is such that the algebraic sum of the perpendiculars let fall upon it from any number of fixed points is zero ; show that it always passes through a fixed point.

If a straight lines moves so that the sum of the perpendiculars let fall on it from the fixed points (3, 4) and (7, 2) is equal to three times the perpendicular on it from a fixed point (1, 3), prove that there is another fixed point through which the line always passes and find its coordinates.

Find the position of the centre of the circle circumscribing the triangle whose vertices are the points (2, 3), (3, 4) and (6, 8).

Find the area of the triangle formed by the straight lines whose equations are 2y + x – 5 = 0, y + 2x – 7 = 0, x – y + 1 = 0.

Find the equation of the straight line which passes through  and perpendicular to the straight line 3x + 4y + 5 = 0

Find the equation of the line through the point (xʹ, yʹ) and perpendicular to the line yyʹ = 2a(x + xʹ)

Find the equation of the straight line which passes through a given point (xʹ, yʹ) and makes a given angle α with the given straight line y = mx + c

Find the equation to the straight line drawn at right angles to the straight line  through the point where it meets the axis of x.

Find the equation of the straight line which bisects, and is perpendicular to, the straight line passing through (a, b) and (aʹ, bʹ).

Prove that the equation of the straight line which passes through the point  and is perpendicular to the straight line  is

Through the point (3, 4) are drawn two straight lines each inclined at  to the straight line x − y = 2. Find their equations and find the area included by the three lines.