A hollow cone is cut by a plane parallel to the base and the upper portion is removed. If the curved surface area of the remainder is $\frac{8}{9}^{th}$ of the curved surface of the whole cone, the ratio of the line segments into which the cone's altitude is divided by the plane is given by

Question

A hollow cone is cut by a plane parallel to the base and the upper portion is removed. If the curved surface area of the remainder is $\frac{8}{9}^{th}$  of the curved surface of the whole cone, the ratio of the line segments into which the cone's altitude is divided by the plane is given by

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Let R and r be the radius and $l$ 1 and  $l$ 2 be slant height of bigger and smaller cone respectively. Curved surface area of cone =  $\pi$ R $($   $l$ 1 +   $l$ 2   $)$ Curved surface Area of remainder =  $\pi$ $($ R + r $)$   $l$ 2   According to question $\frac{8}{9}$ × $\pi$ R $($   $l$ 1 +   $l$ 2   $)$ = $\pi$ $($ R + r $)$   $l$ 2 8R  $l$ 1 =  $l$ 2   $($ R + 9r $)$ $\frac{l_1}{l_2}=\frac{R + 9r}{8R}=$ $($ $\frac{1}{8}$ $+$ $\frac{9r}{8R}$ $)$ ...........(i) According to sin rule = $\frac{R}{l_1 + l_2}=\frac{r}{l_1}$ $\frac{r}{R} = \frac{l_1}{l_1 + l_2}$ .........from eq(i) $\frac{l_1}{l_2}=($ $\frac{1}{8} + \frac{9}{8}$ × $\frac{l_1}{l_1 + l_2}$ $)$ ⇒ $\frac{8l_1}{l_2} - \frac{9l_1}{l_1 + l_2}=1$ $8l_1^2 + 8l_1l_2-9l_1l_2=l_1l_2+l_2^2$   $8l_1^2 - 4l_1l_2+2l_1l_2-l_2^2$ $4l_1(2l_1-l_2)+l_2(2l_1-l_2)=0$ $(2l_1-l_2)(4l_1 + l_2)=0$ $2l_1=l_2$ $\frac{l_1}{l_2}=\frac{1}{2}$ So the correct option is 1:2

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A hollow cone is cut by a plane parallel to the base and the upper portion is removed. If the curved surface area of the remainder is \(\frac{8}{9}^{th}\) of the curved surface of the whole cone, the ratio of the line segments into which the cone's altitude is divided by the plane is given by

The Correct Option is C

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