If the squares of the lengths of the tangents | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let the point be $P(x_1, y_1)$. The lengths of the tangents from $P$ to the circles $x^{2} + y^{2} = a^{2}$,$x^{2} + y^{2} = b^{2}$,and $x^{2} + y

Vedclass · Accepted Answer

$a^{2}, b^{2}, c^{2}$ are in arithmetic progression

Vedclass · Answer

(C) Let the point be $P(x_1, y_1)$. The lengths of the tangents from $P$ to the circles $x^{2} + y^{2} = a^{2}$,$x^{2} + y^{2} = b^{2}$,and $x^{2} + y^{2} = c^{2}$ are given by:$PT_1 = \sqrt{x_1^2 + y_1^2 - a^2}$$PT_2 = \sqrt{x_1^2 + y_1^2 - b^2}$$PT_3 = \sqrt{x_1^2 + y_1^2 - c^2}$Given that $PT_1^2, PT_2^2, PT_3^2$ are in arithmetic progression $(AP)$,we have:$2PT_2^2 = PT_1^2 + PT_3^2$Substituting the values:$2(x_1^2 + y_1^2 - b^2) = (x_1^2 + y_1^2 - a^2) + (x_1^2 + y_1^2 - c^2)$$2x_1^2 + 2y_1^2 - 2b^2 = 2x_1^2 + 2y_1^2 - a^2 - c^2$$-2b^2 = -a^2 - c^2$$2b^2 = a^2 + c^2$This implies that $a^2, b^2, c^2$ are in arithmetic progression.

If the squares of the lengths of the tangents drawn from a point $P$ to the circles $x^{2} + y^{2} = a^2$,$x^2 + y^{2} = b^2$,and $x^{2} + y^{2} = c^{2}$ are in arithmetic progression,then:

Similar Questions

Two tangents $PQ$ and $PR$ are drawn to the circle $x^2 + y^2 - 2x - 4y - 20 = 0$ from the point $P(16, 7)$. If the centre of the circle is $C$,then the area of quadrilateral $PQCR$ is ............ $sq. \text{ units}$.

If the point $(2, \lambda)$ lies inside the circles $x^2+y^2=13$ and $x^2+y^2+x-2y=14$,then $\lambda$ lies in the set

Two chords of the circle $x^2+y^2-2gx-2hy+g^2+h^2-c^2=0$ pass through the point $(g, h+c)$,and the line $y=x$ bisects these two chords. Then:

If the lines $2x + y + 12 = 0$ and $kx - 3y - 10 = 0$ are conjugate with respect to the circle $x^2 + y^2 - 4x + 3y - 1 = 0$,then $k =$

Suppose $O(0,0)$ is the origin and the line $L = x + y - \lambda = 0$ meets the curve $x^2 + y^2 - 2x - 4y + 2 = 0$ at $A$ and $B$. If $\angle AOB = 90^{\circ}$,then the distance between such lines $L = 0$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module